Finite Weyl groupoids of rank three

arXiv:0912.0212v1 [math.GR] 1 Dec 2009

FINITE WEYL GROUPOIDS OF RANK THREE M. CUNTZ AND I. HECKENBERGER Abstract. We continue our study of Cartan schemes and their Weyl groupoids. The results in this paper provide an algorithm to determine connected simply connected Cartan schemes of rank three, where the real roots form a finite irreducible root system. The algorithm terminates: Up to equivalence there are exactly 55 such Cartan schemes, and the number of corresponding real roots varies between 6 and 37. We identify those Weyl groupoids which appear in the classification of Nichols algebras of diagonal type.

1. Introduction Root systems associated with Cartan matrices are widely studied structures in many areas of mathematics, see [Bou68] for the fundaments. The origins of the theory of root systems go back at least to the study of Lie groups by Lie, Killing and Cartan. The symmetry of the root system is commonly known as its Weyl group. Root systems associated with a family of Cartan matrices appeared first in connection with Lie superalgebras [Kac77, Prop. 2.5.6] and with Nichols algebras [Hec06], [Hec08]. The corresponding symmetry is not a group but a groupoid, and is called the Weyl groupoid of the root system. Weyl groupoids of root systems properly generalize Weyl groups. The nice properties of this more general structure have been the main motivation to develop an axiomatic approach to the theory, see [HY08], [CH09c]. In particular, Weyl groupoids are generated by reflections and Coxeter relations, and they satisfy a Matsumoto type theorem [HY08]. To see more clearly the extent of generality it would be desirable to have a classification of finite Weyl groupoids.1 However, already the appearance of a large family of examples of Lie superalgebras and Nichols algebras of diagonal type indicated that a classification of finite Weyl groupoids is probably much more complicated than the classification I.H. is supported by the German Research Foundation (DFG) via a Heisenberg fellowship. 1 In this introduction by a Weyl groupoid we will mean the Weyl groupoid of a connected Cartan scheme, and we assume that the real roots associated to the Weyl groupoid form an irreducible root system in the sense of [CH09c]. 1

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of finite Weyl groups. Additionally, many of the usual classification tools are not available in this context because of the lack of the adjoint action and a positive definite bilinear form. In previous work, see [CH09b] and [CH09a], we have been able to determine all finite Weyl groupoids of rank two. The result of this classification is surprisingly nice: We found a close relationship to the theory of continued fractions and to cluster algebras of type A. The structure of finite rank two Weyl groupoids and the associated root systems has a natural characterization in terms of triangulations of convex polygons by non-intersecting diagonals. In particular, there are infinitely many such groupoids. At first view there is no reason to assume that the situation for finite Weyl groupoids of rank three would be much different from the rank two case. In this paper we give some theoretical indications which strengthen the opposite point of view. For example in Theorem 3.13 we show that the entries of the Cartan matrices in a finite Weyl groupoid cannot be smaller than −7. Recall that for Weyl groupoids there is no lower bound for the possible entries of generalized Cartan matrices. Our main achievement in this paper is to provide an algorithm to classify finite Weyl groupoids of rank three. Our algorithm terminates within a short time, and produces a finite list of examples. In the appendix we list the root systems characterizing the Weyl groupoids of the classification: There are 55 of them which correspond to pairwise non-isomorphic Weyl groupoids. The number of positive roots in these root systems varies between 6 and 37. Among our root systems are the usual root systems of type A3 , B3 , and C3 , but for most of the other examples we don’t have yet an explanation. It is remarkable that the number 37 has a particular meaning for simplicial arrangements in the real projective plane. An arrangement is the complex generated by a family of straight lines not forming a pencil. The vertices of the complex are the intersection points of the lines, the edges are the segments of the lines between two vertices, and the faces are the connected components of the complement of the set of lines generating the arrangement. An arrangement is called simplicial, if all faces are triangles. Simplicial arrangements have been introduced in [Mel41]. The classification of simplicial arrangements in the real projective plane is an open problem. The largest known exceptional example is generated by 37 lines. Gr¨ unbaum conjectures that the list given in [Gr¨ u09] is complete. In our appendix we provide some data of our root systems which can be used to compare Gr¨ unbaum’s list with Weyl groupoids. There is an astonishing analogy between the two lists,

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but more work has to be done to be able to explain the precise relationship. This would be desirable in particular since our classification of finite Weyl groupoids of rank three does not give any evidence for the range of solutions besides the explicit computer calculation. In order to ensure the termination of our algorithm, besides Theorem 3.13 we use a weak convexity property of certain affine hyperplanes, see Theorem 3.11: We can show that any positive root in an affine hyperplane “next to the origin” is either simple or can can be written as the sum of a simple root and another positive root. Our algorithm finally becomes practicable by the use of Proposition 3.6, which can be interpreted as another weak convexity property for affine hyperplanes. It is hard to say which of these theorems are the most valuable because avoiding any of them makes the algorithm impracticable (unless one has some replacement). The paper is organized as follows. We start with two sections proving the necessary theorems to formulate the algorithm: The results which do not require that the rank is three are in Section 2, the obstructions for rank three in Section 3. We then describe the algorithm in the next section. Finally we summarize the resulting data and make some observations in the last section. Acknowledgement. We would like to thank B. M¨ uhlherr for pointing out to us the importance of the number 37 for simplicial arrangements in the real projective plane. 2. Cartan schemes and Weyl groupoids We mainly follow the notation in [CH09c, CH09b]. The fundaments of the general theory have been developed in [HY08] using a somewhat different terminology. Let us start by recalling the main definitions. Let I be a non-empty finite set and {αi | i ∈ I} the standard basis of ZI . By [Kac90, §1.1] a generalized Cartan matrix C = (cij )i,j∈I is a matrix in ZI×I such that (M1) cii = 2 and cjk ≤ 0 for all i, j, k ∈ I with j 6= k, (M2) if i, j ∈ I and cij = 0, then cji = 0. Let A be a non-empty set, ρi : A → A a map for all i ∈ I, and C a = (cajk )j,k∈I a generalized Cartan matrix in ZI×I for all a ∈ A. The quadruple C = C(I, A, (ρi )i∈I , (C a )a∈A ) is called a Cartan scheme if (C1) ρ2i = id for all i ∈ I,

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(C2) caij = ciji

for all a ∈ A and i, j ∈ I.

Let C = C(I, A, (ρi )i∈I , (C a )a∈A ) be a Cartan scheme. For all i ∈ I and a ∈ A define σia ∈ Aut(ZI ) by (2.1)

σia (αj ) = αj − caij αi

for all j ∈ I.

The Weyl groupoid of C is the category W(C) such that Ob(W(C)) = A and the morphisms are compositions of maps σia with i ∈ I and a ∈ A, where σia is considered as an element in Hom(a, ρi (a)). The category W(C) is a groupoid in the sense that all morphisms are isomorphisms. The set of morphisms of W(C) is denoted by Hom(W(C)), and we use the notation Hom(a, W(C)) = ∪ Hom(a, b) (disjoint union). b∈A

For notational convenience we will often neglect upper indices referring to elements of A if they are uniquely determined by the context. For ρik (a) a ρi ···ρi (a) σik ∈ Hom(a, b), where k ∈ example, the morphism σi1 2 k · · · σik−1 N, i1 , . . . , ik ∈ I, and b = ρi1 · · · ρik (a), will be denoted by σi1 · · · σiak or by idb σi1 · · · σik . The cardinality of I is termed the rank of W(C). A Cartan scheme is called connected if its Weyl groupoid is connected, that is, if for all a, b ∈ A there exists w ∈ Hom(a, b). The Cartan scheme is called simply connected, if Hom(a, a) = {ida } for all a ∈ A. Let C be a Cartan scheme. For all a ∈ A let (Rre )a = {ida σi1 · · · σik (αj ) | k ∈ N0 , i1 , . . . , ik , j ∈ I} ⊂ ZI . The elements of the set (Rre )a are called real roots (at a). The pair (C, ((Rre)a )a∈A ) is denoted by Rre (C). A real root α ∈ (Rre )a , where a ∈ A, is called positive (resp. negative) if α ∈ NI0 (resp. α ∈ −NI0 ). In contrast to real roots associated to a single generalized Cartan matrix, (Rre )a may contain elements which are neither positive nor negative. A good general theory, which is relevant for example for the study of Nichols algebras, can be obtained if (Rre )a satisfies additional properties. Let C = C(I, A, (ρi )i∈I , (C a )a∈A ) be a Cartan scheme. For all a ∈ A let Ra ⊂ ZI , and define mai,j = |Ra ∩ (N0 αi + N0 αj )| for all i, j ∈ I and a ∈ A. We say that R = R(C, (Ra )a∈A ) is a root system of type C, if it satisfies the following axioms. a a a (R1) Ra = R+ ∪ −R+ , where R+ = Ra ∩ NI0 , for all a ∈ A. a (R2) R ∩ Zαi = {αi , −αi } for all i ∈ I, a ∈ A. (R3) σia (Ra ) = Rρi (a) for all i ∈ I, a ∈ A.

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Figure 1. The object change diagram of a Cartan scheme of rank three (nr. 15 in Table 1)

(R4) If i, j ∈ I and a ∈ A such that i 6= j and mai,j is finite, then a (ρi ρj )mi,j (a) = a. The axioms (R2) and (R3) are always fulfilled for Rre . The root system R is called finite if for all a ∈ A the set Ra is finite. By [CH09c, Prop. 2.12], if R is a finite root system of type C, then R = Rre , and hence Rre is a root system of type C in that case. In [CH09c, Def. 4.3] the concept of an irreducible root system of type C was defined. By [CH09c, Prop. 4.6], if C is a Cartan scheme and R is a finite root system of type C, then R is irreducible if and only if for all a ∈ A the generalized Cartan matrix C a is indecomposable. If C is also connected, then it suffices to require that there exists a ∈ A such that C a is indecomposable. Let C = C(I, A, (ρi )i∈I , (C a )a∈A ) be a Cartan scheme. Let Γ be a nondirected graph, such that the vertices of Γ correspond to the elements of A. Assume that for all i ∈ I and a ∈ A with ρi (a) 6= a there is precisely one edge between the vertices a and ρi (a) with label i, and all edges of Γ are given in this way. The graph Γ is called the object change diagram of C. In the rest of this section let C = C(I, A, (ρi )i∈I , (C a )a∈A ) be a Cartan scheme such that Rre (C) is a finite root system. For brevity we will write Ra instead of (Rre )a for all a ∈ A. We say that a subgroup H ⊂ ZI is a hyperplane if ZI /H ∼ = Z. Then rk H = #I − 1 is the rank of H. Sometimes we will identify ZI with its image under the canonical embedding ZI → Q ⊗Z ZI ∼ = QI .

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Lemma 2.1. Let a ∈ A and let H ⊂ ZI be a hyperplane. Suppose that H contains rk H linearly independent elements of Ra . Let nH be a normal vector of H in QI with respect to a scalar product (·, ·) on QI . a If (nH , α) ≥ 0 for all α ∈ R+ , then H contains rk H simple roots, and all roots contained in H are linear combinations of these simple roots. Proof. The assumptions imply that any positive root in H is a linear a a combination of simple roots contained in H. Since Ra = R+ ∪ −R+ , this implies the claim.  Lemma 2.2. Let a ∈ A and let H ⊂ ZI be a hyperplane. Suppose that H contains rk H linearly independent elements of Ra . Then there exist b ∈ A and w ∈ Hom(a, b) such that w(H) contains rk H simple roots. Proof. Let (·, ·) be a scalar product on QI . Choose a normal vector nH a of H in QI with respect to (·, ·). Let m = #{α ∈ R+ | (nH , α) < 0}. re Since R (C) is finite, m is a nonnegative integer. We proceed by induction on m. If m = 0, then H contains rk H simple roots by Lemma 2.1. Otherwise let j ∈ I with (nH , αj ) < 0. Let a′ = ρj (a) and H ′ = σja (H). Then σja (nH ) is a normal vector of H ′ with respect to ρ (a) ρ (a) a the scalar product (·, ·)′ = (σj j (·), σj j (·)). Since σja : R+ \ {αj } → ′ a a R+ \ {αj } is a bijection and σj (αj ) = −αj , we conclude that ′

a a #{β ∈ R+ | (σja(nH ), β)′ < 0} = #{α ∈ R+ | (nH , α) < 0} − 1.

By induction hypothesis there exists b ∈ A and w ′ ∈ Hom(a′ , b) such that w ′ (H ′ ) contains rk H ′ = rk H simple roots. Then the claim of the lemma holds for w = w ′ σja .  The following “volume” functions will be useful for our analysis. Let k ∈ N with k ≤ #I. By the Smith normal form there is a unique left GL(ZI )-invariant right GL(Zk )-invariant function Volk : (ZI )k → Z such that (2.2)

Volk (a1 α1 , . . . , ak αk ) = |a1 · · · ak | for all a1 , . . . , ak ∈ Z,

where |·| denotes absolute value. In particular, if k = 1 and β ∈ ZI \{0}, then Vol1 (β) is the largest integer v such that β = vβ ′ for some β ′ ∈ ZI . Further, if k = #I and β1 , . . . , βk ∈ ZI , then Volk (β1 , . . . , βk ) is the absolute value of the determinant of the matrix with columns β1 , . . . , βk . Let a ∈ A, k ∈ {1, 2, . . . , #I}, and let β1 , . . . , βk ∈ Ra be linearly independent elements. We write V a (β1 , . . . , βk ) for the unique maximal subgroup V ⊆ ZI of rank k which contains β1 , . . . , βk . Then ZI /V a (β1 , . . . , βk ) is free. In particular, V a (β1 , . . . , β#I ) = ZI for all a ∈ A and any linearly independent subset {β1 , . . . , β#I } of Ra .

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Definition 2.3. Let W ⊆ ZI be a cofree subgroup (that is, ZI /W is free) of rank k. We say that {β1 , . . . , βk } is a base for W at a, if βi ∈ W P P for all i ∈ {1, . . . , k} and W ∩ Ra ⊆ ki=1 N0 βi ∪ − ki=1 N0 βi .

Now we discuss the relationship of linearly independent roots in a root system. Recall that C is a Cartan scheme such that Rre (C) is a finite root system of type C. Theorem 2.4. Let a ∈ A, k ∈ {1, . . . , #I}, and let β1 , . . . , βk ∈ Ra be linearly independent roots. Then there exist b ∈ A, w ∈ Hom(a, b), and a permutation τ of I such that b w(βi) ∈ spanZ {ατ (1) , . . . , ατ (i) } ∩ R+

for all i ∈ {1, . . . , k}. Proof. Let r = #I. Since Ra contains r simple roots, any linearly independent subset of Ra can be enlarged to a linearly independent subset of r elements. Hence it suffices to prove the theorem for k = r. We proceed by induction on r. If r = 1, then the claim holds. Assume that r > 1. Lemma 2.2 with H = V a (β1 , . . . , βr−1 ) tells that there exist d ∈ A and v ∈ Hom(a, d) such that v(H) is spanned by simple roots. By multiplying v from the left with the longest element of W(C) in the case that v(βr ) ∈ −NI0 , we may even assume that v(βr ) ∈ NI0 . Now let J be the subset of I such that #J = r − 1 and αi ∈ v(H) for all i ∈ J. Consider the restriction Rre (C)|J of Rre (C) to the index set J, see [CH09c, Def. 4.1]. Since v(βi ) ∈ H for all i ∈ {1, . . . , r − 1}, induction hypothesis provides us with b ∈ A, u ∈ Hom(d, b), and a permutation τ ′ of J such that u is a product of simple reflections σi , where i ∈ J, and b uv(βn ) ∈ spanZ {ατ ′ (j1 ) , . . . , ατ ′ (jn ) } ∩ R+

for all n ∈ {1, 2, . . . , r − 1}, where J = {j1 , . . . , jr−1 }. Since v(βr ) ∈ / I v(H) and v(βr ) ∈ N0 , the i-th entry of v(βr ), where i ∈ I \J, is positive. This entry does not change if we apply u. Therefore uv(βr ) ∈ NI0 , and hence the theorem holds with w = uv ∈ Hom(a, b) and with τ the unique permutation with τ (n) = τ ′ (jn ) for all n ∈ {1, . . . , r − 1}.  Corollary 2.5. Let a ∈ A, k ∈ {1, . . . , #I}, and let β1 , . . . , βk ∈ Ra be linearly independent elements. Then {β1 , . . . , βk } is a base for V a (β1 , . . . , βk ) at a if and only if there exist b ∈ A, w ∈ Hom(a, b), and a permutation τ of I such that w(βi ) = ατ (i) for all i ∈ {1, . . . , k}. In this case Volk (β1 , . . . , βk ) = 1. Proof. The if part of the claim holds by definition of a base and by the axioms for root systems.

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Let b, w and τ be as in Theorem 2.4. Let i ∈ {1, . . . , k}. The elements w(β1 ), . . . , w(βi) are linearly independent and are contained in V b (ατ (1) , . . . , ατ (i) ). Thus ατ (i) is a rational linear combination of w(β1), . . . , w(βi). Now by assumption, {w(β1 ), . . . , w(βk )} is a base for V b (w(β1 ), . . . , w(βk )) at b. Hence ατ (i) is a linear combination of the positive roots w(β1 ), . . . , w(βi) with nonnegative integer coefficients. This is possible only if {w(β1), . . . , w(βi)} contains ατ (i) . By induction on i we obtain that ατ (i) = w(βi).  In the special case k = #I the above corollary tells that the bases of Z at an object a ∈ A are precisely those subsets which can be obtained as the image, up to a permutation, of the standard basis of ZI under the action of an element of W(C). In [CH09a] the notion of an F -sequence was given, and it was used to explain the structure of root systems of rank two. Consider on N20 the total ordering ≤Q , where (a1 , a2 ) ≤Q (b1 , b2 ) if and only if a1 b2 ≤ a2 b1 . A finite sequence (v1 , . . . , vn ) of vectors in N20 is an F -sequence if and only if v1 0. We may assume that for all j ∈ I, b′ ∈ A, and w ′ ∈ Hom(b′ , a) with w ′ (αj ) = α we have ℓ(w ′) ≥ ℓ(w). Since w(αi ) ∈ NI0 , we obtain that ℓ(wσi ) > ℓ(w) [HY08, Cor. 3]. Therefore, there is a j ∈ I \ {i} with ℓ(wσj ) < ℓ(w). Let w = w1 w2 such that ℓ(w) = ℓ(w1 ) + ℓ(w2 ), ℓ(w1 ) minimal and w2 = . . . σi σj σi σjb . Assume that w2 = σi · · · σi σjb — the case w2 = σj · · · σi σjb can be treated similarly. The length of w1 is minimal, thus ℓ(w1 σj ) > ℓ(w1 ), and ℓ(w) = ℓ(w1 ) + ℓ(w2 ) yields that ℓ(w1 σi ) > ℓ(w1 ). Using once more [HY08, Cor. 3] we conclude that (2.3)

w1 (αi ) ∈ NI0 ,

w1 (αj ) ∈ NI0 .

Let β = w2 (αi ). Then β ∈ N0 αi + N0 αj , since ℓ(w2 σi ) > ℓ(w2 ). Moreover, β is not simple. Indeed, α = w(αi ) = w1 (β), so β is not simple, since ℓ(w1 ) < ℓ(w) and ℓ(w) was chosen of minimal length. By Proposition 2.7 we conclude that β is the sum of two positive roots β1 , β2 ∈ N0 αi + N0 αj . It remains to check that w1 (β1 ), w1 (β2 ) are positive. But this follows from (2.3).  3. Obstructions for Weyl groupoids of rank three In this section we analyze the structure of finite Weyl groupoids of rank three. Let C be a Cartan scheme of rank three, and assume that Rre (C) is a finite irreducible root system of type C. In this case a hyperplane in ZI is the same as a cofree subgroup of rank two, which will be called a plane in the sequel. For simplicity we will take I = {1, 2, 3}, and we write Ra for the set of positive real roots at a ∈ A. Recall the definition of the functions Volk , where k ∈ {1, 2, 3}, from the previous section. As noted, for three elements α, β, γ ∈ Z3 we have Vol3 (α, β, γ) = 1 if and only if {α, β, γ} is a basis of Z3 . Also, we will heavily use the notion of a base, see Definition 2.3. Lemma 3.1. Let a ∈ A and α, β ∈ Ra . Assume that α 6= ±β and that {α, β} is not a base for V a (α, β) at a. Then there exist k, l ∈ N and δ ∈ Ra such that β − kα = lδ and {α, δ} is a base for V a (α, β) at a.

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Proof. The claim without the relation k > 0 is a special case of Theorem 2.4. The relation β 6= δ follows from the assumption that {α, β} is not a base for V a (α, β) at a.  Lemma 3.2. Let a ∈ A and α, β ∈ Ra such that α 6= ±β. Then {α, β} is a base for V a (α, β) if and only if Vol2 (α, β) = 1 and α − β ∈ / Ra . Proof. Assume first that {α, β} is a base for V a (α, β) at a. By Corollary 2.5 we may assume that α and β are simple roots. Therefore Vol2 (α, β) = 1 and α − β ∈ / Ra . Conversely, assume that Vol2 (α, β) = 1, α − β ∈ / Ra , and that {α, β} a is not a base for V (α, β) at a. Let k, l, δ as in Lemma 3.1. Then 1 = Vol2 (α, β) = Vol2 (α, β − kα) = lVol2 (α, δ). Hence l = 1, and {α, δ} = {α, β −kα} is a base for V a (α, β) at a. Then β − α ∈ Ra by Corollary 2.8 and since k > 0. This gives the desired contradiction to the assumption α − β ∈ / Ra .  Recall that a semigroup ordering < on a commutative semigroup (S, +) is a total ordering such that for all s, t, u ∈ S with s < t the relations s + u < t + u hold. For example, the lexicographic ordering on ZI induced by any total ordering on I is a semigroup ordering. Lemma 3.3. Let a ∈ A, and let V ⊂ ZI be a plane containing at least two positive roots of Ra . Let < be a semigroup ordering on ZI such that a 0 < γ for all γ ∈ R+ , and let α, β denote the two smallest elements in a V ∩ R+ with respect to 0. Since β > 0, we conclude that β = kα + lδ for some k, l ∈ N0 . Since β is not a multiple of α, this implies that β = δ or β > δ. Then the choice of β and the positivity of δ yield that δ = β, that is, {α, β} is a base for V at a.  a Lemma 3.4. Let k ∈ N≥2 , a ∈ A, α ∈ R+ , and β ∈ ZI such that α and β are not collinear and α + kβ ∈ Ra . Assume that Vol2 (α, β) = 1

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and that (−Nα + Zβ) ∩ NI0 = ∅. Then β ∈ Ra and α + lβ ∈ Ra for all l ∈ {1, 2, . . . , k}. Proof. We prove the claim indirectly. Assume that β ∈ / Ra . By a Lemma 3.3 there exists a base {γ1 , γ2 } for V (α, β) at a such that a γ1 , γ2 ∈ R+ . The assumptions of the lemma imply that there exist m1 , l1 ∈ N0 and m2 , l2 ∈ Z such that γ1 = m1 α + m2 β, γ2 = l1 α + l2 β. Since β ∈ / Ra , we obtain that m1 ≥ 1 and m2 ≥ 1. Therefore relaa tions α, α + kβ ∈ R+ imply that {α, α + kβ} = {γ1, γ2 }. The latter is a contradiction to Vol2 (γ1 , γ2 ) = 1 and Vol2 (α, α + kβ) = k > 1. Thus β ∈ Ra . By Lemma 3.1 we obtain that {β, α − mβ} is a base for V a (α, β) at a for some m ∈ N0 . Then Corollary 2.8 and the assumption that α + kβ ∈ Ra imply the last claim of the lemma.  We say that a subset S of Z3 is convex, if any rational convex linear combination of elements of S is either in S or not in Z3 . We start with a simple example. Lemma 3.5. Let a ∈ A. Assume that ca12 = 0. (1) Let k1 , k2 ∈ Z. Then α3 + k1 α1 + k2 α2 ∈ Ra if and only if 0 ≤ k1 ≤ −ca13 and 0 ≤ k2 ≤ −ca23 . (2) Let γ ∈ (α3 + Zα1 + Zα2 ) ∩Ra . Then γ −α1 ∈ Ra or γ + α1 ∈ Ra . Similarly γ − α2 ∈ Ra or γ + α2 ∈ Ra . ρ (a)

Proof. (1) The assumption ca12 = 0 implies that c231 = ca23 , see [CH09c, ρ (a) ρ (a) ρ ρ (a) Lemma 4.5]. Applying σ1 1 , σ2 2 , and σ1 σ2 2 1 to α3 we conclude a that α3 − ca13 α1 , α3 − ca23 α2 , α3 − ca13 α1 − ca23 α2 ∈ R+ . Thus Lemma 3.4 a implies that α3 + m1 α1 + m2 α2 ∈ R for all m1 , m2 ∈ Z with 0 ≤ m1 ≤ −ca13 and 0 ≤ m2 ≤ −ca23 . Further, (R1) gives that α3 +k1 α1 +k2 α2 ∈ / Ra if k1 < 0 or k2 < 0. Applying again the simple reflections σ1 and σ2 , a similar argument proves the remaining part of the claim. Observe that the proof does not use the fact that Rre (C) is irreducible. (2) Since ca12 = 0, the irreducibility of Rre (C) yields that ca13 , ca23 < 0 by [CH09c, Def. 4.5, Prop. 4.6]. Hence the claim follows from (1).  Proposition 3.6. Let a ∈ A and let γ1 , γ2 , γ3 ∈ Ra . Assume that Vol3 (γ1 , γ2 , γ3) = 1 and that γ1 − γ2 , γ1 − γ3 ∈ / Ra . Then γ1 + γ2 ∈ Ra a or γ1 + γ3 ∈ R . Proof. Since γ1 − γ2 ∈ / Ra and Vol3 (γ1 , γ2 , γ3) = 1, Theorem 2.4 and Lemma 3.2 imply that there exists b ∈ A, w ∈ Hom(a, b) and i1 , i2 , i3 ∈ I such that w(γ1 ) = αi1 , w(γ2) = αi2 , and w(γ3) = αi3 + k1 αi1 + k2 αi2 for some k1 , k2 ∈ N0 . Assume that γ1 + γ2 ∈ / Ra . Then cbi1 i2 = 0. Since a γ3 − γ1 ∈ / R , Lemma 3.5(2) with γ = w(γ3 ) gives that γ3 + γ1 ∈ Ra . This proves the claim. 

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Lemma 3.7. Assume that Ra ∩ (N0 α1 + N0 α2 ) contains at most 4 positive roots. (1) The set S3 := (α3 + Zα1 + Zα2 ) ∩ Ra is convex. (2) Let γ ∈ S3 . Then γ = α3 or γ − α1 ∈ Ra or γ − α2 ∈ Ra . Proof. Consider the roots of the form w −1 (α3 ) ∈ Ra , where w ∈ Hom(a, W(C)) is a product of reflections σ1b , σ2b with b ∈ A. All of these roots belong to S3 . Using Lemma 3.4 the claims of the lemma can be checked case by case, similarly to the proof of Lemma 3.5.  Remark 3.8. The lemma can be proven by elementary calculations, since all nonsimple positive roots in (Zα1 + Zα2 ) ∩ Ra are of the form say α1 + kα2 , k ∈ N. We will see in Theorem 4.1 that the classification of connected Cartan schemes of rank three admitting a finite irreducible root system has a finite set of solutions. Thus it is possible to check the claim of the lemma for any such Cartan scheme. Using computer calculations one obtains that the lemma holds without any restriction on the (finite) cardinality of Ra ∩ (N0 α1 + N0 α2 ). Lemma 3.9. Let α, β, γ ∈ Ra such that Vol3 (α, β, γ) = 1. Assume that α − β, β − γ, α − γ ∈ / Ra and that {α, β, γ} is not a base for ZI at a. Then the following hold. (1) There exist w ∈ Hom(a, W(C)) and n1 , n2 ∈ N such that w(α), w(β), and w(γ − n1 α − n2 β) are simple roots. (2) None of the vectors α − kβ, α − kγ, β − kα, β − kγ, γ − kα, γ − kβ, where k ∈ N, is contained in Ra . (3) α + β, α + γ, β + γ ∈ Ra . (4) One of the sets {α+2β, β+2γ, γ+2α} and {2α+β, 2β+γ, 2γ+α} is contained in Ra , the other one has trivial intersection with Ra . (5) None of the vectors γ −α−kβ, γ −kα−β, β −γ −kα, β −kγ −α, α − β − kγ, α − kβ − γ, where k ∈ N0 , is contained in Ra . (6) Assume that α + 2β ∈ Ra . Let k ∈ N such that α + kβ ∈ Ra , α + (k + 1)β ∈ / Ra . Let α′ = α + kβ, β ′ = −β, γ ′ = γ + β. Then Vol3 (α′ , β ′, γ ′ ) = 1, {α′ , β ′, γ ′ } is not a base for ZI at a, and none of α′ − β ′ , α′ − γ ′ , β ′ − γ ′ is contained in Ra . (7) None of the vectors α + 3β, β + 3γ, γ + 3α, 3α + β, 3β + γ, 3γ + α is contained in Ra . In particular, k = 2 holds in (6). Proof. (1) By Theorem 2.4 there exist m1 , m2 , n1 , n2 , n3 ∈ N0 , i1 , i2 , i3 ∈ I, and w ∈ Hom(a, W(C)), such that w(α) = αi1 , w(β) = m1 αi1 + m2 αi2 , and w(γ) = n1 αi1 + n2 αi2 + n3 αi3 . Since det w ∈ {±1} and Vol3 (α, β, γ) = 1, this implies that m2 = n3 = 1. Further, β − α ∈ / Ra , and hence w(β) = αi2 by Corollary 2.9. Since {α, β, γ} is not a base for ZI at a, we conclude that w(γ) 6= αi3 . Then Corollary 2.9 and the

FINITE WEYL GROUPOIDS OF RANK THREE

13

assumptions γ − α, γ − β ∈ / Ra imply that w(γ) ∈ / αi3 + N0 αi1 and w(γ) ∈ / αi3 + N0 αi2 . Thus the claim is proven. (2) By (1), {α, β} is a base for V a (α, β) at a. Thus α − kβ ∈ / Ra for all k ∈ N. The remaining claims follow by symmetry. (3) Suppose that α + β ∈ / Ra . By (1) there exist b ∈ A, w ∈ Hom(a, b), i1 , i2 , i3 ∈ I and n1 , n2 ∈ N such that w(α) = αi1 , w(β) = b αi2 , and w(γ) = αi3 + n1 αi1 + n2 αi2 ∈ R+ . By Theorem 2.10 there exist ′ ′ ′ ′ n1 , n2 ∈ N0 such that n1 ≤ n1 , n2 ≤ n2 , n′1 + n′2 < n1 + n2 , and b αi3 + n′1 αi1 + n′2 αi2 ∈ R+ ,

b (n1 − n′1 )αi1 + (n2 − n′2 )αi2 ∈ R+ .

b Since α + β ∈ / Ra , Proposition 2.6 yields that R+ ∩ spanZ {αi1 , αi2 } = a a {αi1 , αi2 }. Thus γ − α ∈ R or γ − β ∈ R . This is a contradiction to the assumption of the lemma. Hence α + β ∈ Ra . By symmetry we obtain that α + γ, β + γ ∈ Ra . (4) Suppose that α + 2β, 2α + β ∈ / Ra . By (1) the set {α, β} is a base for V a (α, β) at a, and α + β ∈ Ra by (3). Then Proposition 2.6 implies that Ra ∩ spanZ {α, β} = {±α, ±β, ±(α + β)}. Thus (1) and Lemma 3.7(2) give that γ − α ∈ Ra or γ − β ∈ Ra , a contradiction to the initial assumption of the lemma. Hence by symmetry each of the sets {α + 2β, 2α + β}, {α + 2γ, 2α + γ}, {β + 2γ, 2β + γ} contains at least one element of Ra . Assume now that γ + 2α, γ + 2β ∈ Ra . By changing the object via (1) we may assume that α, β, and γ − n1 α − n2 β are simple roots for a some n1 , n2 ∈ N. Then Lemma 3.4 applies to γ + 2α ∈ R+ and β − α, a and tells that β − α ∈ R . This gives a contradiction. By the previous two paragraphs we conclude that if γ + 2α ∈ Ra , then γ + 2β ∈ / Ra , and hence β + 2γ ∈ Ra . Similarly, we also obtain that α + 2β ∈ Ra . By symmetry this implies (4). (5) By symmetry it suffices to prove that γ − (α + kβ) ∈ / Ra for all k ∈ N0 . For k = 0 the claim holds by assumption. First we prove that γ − (α + 2β) ∈ / Ra . By (3) we know that γ + α, a a α+β ∈ R , and γ −β ∈ / R by assumption. Since Vol2 (γ +α, α+β) = 1, Lemma 3.2 gives that {γ + α, α + β} is a base for V a (γ + α, α + β) at a. Since γ − (α + 2β) = (γ + α) − 2(α + β), we conclude that γ − (α + 2β) ∈ / Ra . Now let k ∈ N. Assume that γ − (α + kβ) ∈ Ra and that k is minimal with this property. Let α′ = −α, β ′ = −β, γ ′ = γ − (α + kβ). Then α′ , β ′ , γ ′ ∈ Ra with Vol3 (α′ , β ′ , γ ′ ) = 1. Moreover, α′ − β ′ ∈ / Ra by assumption, α′ − γ ′ = −(γ − kβ) ∈ / Ra by (2), and β ′ − γ ′ = a −(γ − α − (k − 1)β) ∈ / R by the minimality of k. Further, {α′ , β ′ , γ ′ } is not a base for Ra , since γ = γ ′ − α′ − kβ ′ . Hence Claim (3) holds for

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α′ , β ′, γ ′ . In particular, γ ′ + β ′ = γ − (α + (k + 1)β) ∈ Ra . This and the previous paragraph imply that k ≥ 3. We distinguish two cases depending on the parity of k. First assume that k is even. Let α′ = γ + α and β ′ = −(α + k/2β). Then Vol2 (α′ , β ′) = 1 and α′ + 2β ′ = γ − (α + kβ) ∈ Ra . Lemma 3.4 applied to α′ , β ′ gives that γ − k/2β = α′ + β ′ ∈ Ra , which contradicts (2). Finally, the case of odd k can be excluded similarly by considering V a (γ + α, γ − (α + (k + 1)β)). (6) We get Vol3 (α′ , β ′ , γ ′ ) = 1 since Vol3 (α, β, γ) = 1 and Vol3 is invariant under the right action of GL(Z3 ). Further, β ′ − γ ′ = −(2β + γ) ∈ / Ra by (4), and α′ − γ ′ ∈ / Ra by (5). Finally, (α′ , β ′, γ ′ ) is not a I a base for Z at a, since R ∋ γ −n1 α−n2 β = γ ′ −n1 α′ +(1+n2 −kn1 )β ′ , where n1 , n2 ∈ N are as in (1). (7) We prove that γ + 3α ∈ / Ra . The rest follows by symmetry. If 2α + β ∈ Ra , then γ + 2α ∈ / Ra by (4), and hence γ + 3α ∈ / Ra . a ′ ′ ′ Otherwise α + 2β, γ + 2α ∈ R by (4). Let k, α , β , γ be as in (6). Then (6) and (3) give that Ra ∋ γ ′ + α′ = γ + α + (k + 1)β. Since γ + α ∈ Ra , Lemma 3.4 implies that γ + α + 2β ∈ Ra . Let w be as in (1). If γ + 3α ∈ Ra , then Lemma 3.4 for the vectors w(γ + α + 2β) and w(α − β) implies that w(α − β) ∈ Ra , a contradiction. Thus γ + 3α ∈ / Ra .  Recall that C is a Cartan scheme of rank three and Rre (C) is a finite irreducible root system of type C. Theorem 3.10. Let a ∈ A and α, β, γ ∈ Ra . If Vol3 (α, β, γ) = 1 and none of α − β, α − γ, β − γ are contained in Ra , then {α, β, γ} is a base for ZI at a. Proof. Assume to the contrary that {α, β, γ} is not a base for ZI at a. Exchanging α and β if necessary, by Lemma 3.9(4) we may assume that α + 2β ∈ Ra . By Lemma 3.9(6),(7) the triple (α + 2β, −β, γ + β) satisfies the assumptions of Lemma 3.9, and (α+2β)+2(−β) = α ∈ Ra . Hence 2α + 3β = 2(α + 2β) + (−β) ∈ / Ra by Lemma 3.9(4). Thus a a V (α, β) ∩ R = {±α, ±(α + β), ±(α + 2β), ±β} by Proposition 2.6, and hence, using Lemma 3.9(1), we obtain from Lemma 3.7(2) that γ − α ∈ Ra or γ − β ∈ Ra . This is a contradiction to our initial assumption, and hence {α, β, γ} is a base for ZI at a.  Corollary 3.11. Let a ∈ A and γ1 , γ2, α ∈ Ra . Assume that {γ1 , γ2 } is a base for V a (γ1 , γ2 ) at a and that Vol3 (γ1 , γ2, α) = 1. Then either

FINITE WEYL GROUPOIDS OF RANK THREE

15

{γ1, γ2 , α} is a base for ZI at a, or one of α − γ1 , α − γ2 is contained in Ra . For the proof of Theorem 4.1 we need a bound for the entries of the Cartan matrices of C. To get this bound we use the following. Lemma 3.12. Let a ∈ A. (1) At most one of ca12 , ca13 , ca23 is zero. (2) α1 + α2 + α3 ∈ Ra . (3) Let k ∈ Z. Then kα1 + α2 + α3 ∈ Ra if and only if k1 ≤ k ≤ k2 , where ( ( ρ (a) 0 if ca23 < 0, −ca12 − ca13 if c231 < 0, k1 = k2 = ρ (a) 1 if ca23 = 0, −ca12 − ca13 − 1 if c231 = 0. (4) We have 2α1 + α2 + α3 ∈ Ra if and only if either ca12 + ca13 ≤ −3 ρ (a) or ca12 + ca13 = −2, c231 < 0. (5) Assume that (3.1)

a ∩ (Zα1 + Zα2 )) ≥ 5. #(R+

Then there exist k ∈ N0 such that kα1 + 2α2 + α3 ∈ Ra . Let k0 be the smallest among all such k. Then k0 is given by the following.   0 if ca23 ≤ −2,    ρ2 (a) a a a   1 if −1 ≤ c23 ≤ 0, c21 + c23 ≤ −2, c13 < 0,   ρ (a)  1 if −1 ≤ ca23 ≤ 0, ca21 + ca23 ≤ −3, c132 = 0,    2 if ca = ca = −1, cρ2 (a) = 0, 23 13 21 ρ (a) a a  2 if c = −1, c = 0, c132 ≤ −2,  21 23   ρ2 (a) ρ2 (a) a a   3 if c21 = −1, c23 = 0, c13 = −1, c12 ≤ −3,   ρ (a) ρ (a) ρ ρ (a)  3 if ca21 = −1, ca23 = 0, c132 = −1, c122 = −2, c231 2 < 0,    4 otherwise. Further, if ca13 = 0 then k0 ≤ 2.

Proof. We may assume that C is connected. Then, since Rre (C) is irreducible, Claim (1) holds by [CH09c, Def. 4.5, Prop. 4.6]. (2) The claim is invariant under permutation of I. Thus by (1) we may assume that ca23 6= 0. Hence α2 + α3 ∈ Ra . Assume first that ρ (a) ρ (a) ρ (a) ca13 = 0. Then c131 = 0 by (C2), c231 6= 0 by (1), and α2 +α3 ∈ R+1 . ρ (a) Hence σ1 1 (α2 + α3 ) = −ca12 α1 + α2 + α3 ∈ Ra . Therefore (2) holds by Lemma 3.4 for α = α2 + α3 and β = α1 . Assume now that ca13 6= 0. By symmetry and the previous paragraph we may also assume that ca12 , ca23 6= 0. Let b = ρ1 (a). If cb23 = 0 then

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α1 + α2 + α3 ∈ Rb by the previous paragraph. Then Ra ∋ σ1b (α1 + α2 + α3 ) = (−ca12 − ca13 − 1)α1 + α2 + α3 , and the coefficient of α1 is positive. Further, α2 + α3 ∈ Ra , and hence (2) holds in this case by Lemma 3.4. Finally, if cb23 6= 0, then α2 + α3 ∈ b R+ , and hence (−ca12 − ca13 )α1 + α2 + α3 ∈ Ra . Since −ca12 − ca13 > 0, (2) follows again from Lemma 3.4. (3) If ca23 < 0, then α2 + α3 ∈ Ra and −kα1 + α2 + α3 ∈ / Ra for all a a k ∈ N. If c23 = 0, then α1 +α2 +α3 ∈ R by (2), and −kα1 +α2 +α3 ∈ / Ra for all k ∈ N0 . Applying the same argument to Rρ1 (a) and using the ρ (a) reflection σ1 1 and Lemma 3.4 gives the claim. (4) This follows immediately from (3). (5) The first case follows from Corollary 2.9 and the second and third cases are obtained from (4) by interchanging the elements 1 and 2 of I. We also obtain that if k0 exists then k0 ≥ 2 in all other cases. By (3.1) and Proposition 2.6 we conclude that α1 + α2 ∈ Ra . Then ca21 < 0 by Corollary 2.9, and hence we are left with calculating k0 if −1 ≤ ca23 ≤ 0, ρ (a) ρ (a) ca21 + ca23 = −2, c132 = 0, or ca21 = −1, ca23 = 0. By (1), if c132 = 0 ρ (a) then c232 6= 0, and hence ca23 < 0 by (C2). Thus we have to consider the elements kα1 + 2α2 + α3 , where k ≥ 2, under the assumption that (3.2)

ρ (a)

ca21 = ca23 = −1, c132

= 0 or ca21 = −1, ca23 = 0.

Since ca21 = −1, Condition (3.1) gives that ρ (a)

c122

≤ −2,

see [CH09c, Lemma 4.8]. Further, the first set of equations in (3.2) ρ ρ (a) ρ ρ (a) implies that c131 2 = 0, and hence c231 2 < 0 by (1). Since σ2a (2α1 + 2α2 + α3 ) = 2α1 + α3 − ca23 α2 , the first set of equations in (3.2) and (4) imply that k0 = 2. Similarly, Corollary 2.9 tells that k0 = 2 under the ρ (a) second set of conditions in (3.2) if and only if c132 ≤ −2. It remains to consider the situation for (3.3)

ρ (a)

ca21 = −1, ca23 = 0, c132

= −1.

ρ (a)

Indeed, equation ca23 = 0 implies that c232 = 0 by (C2), and hence ρ (a) c132 < 0 by (1), Assuming (3.3) we obtain that σ2a (3α1 + 2α2 + α3 ) = 3α1 + α2 + α3 , and hence (3) implies that k0 = 3 if and only if the corresponding conditions in (5) are valid. The rest follows by looking at σ1 σ2a (4α1 + 2α2 + α3 ) and is left to the reader. The last claim holds since ca13 = 0 implies that ca23 6= 0 by (1). a The assumption #(R+ ∩ (Zα1 + Zα2 )) ≥ 5 is needed to exclude the ρ (a) ρ ρ (a) 2 a a case c21 = −1, c12 = −2, c211 2 = −1, where R+ ∩ (Zα1 + Zα2 ) =

FINITE WEYL GROUPOIDS OF RANK THREE

17

{α2 , α1 + α2 , 2α1 + α2 , α1 }, by using Proposition 2.6 and Corollary 2.9, see also the proof of [CH09c, Lemma 4.8].  Theorem 3.13. Let C be a Cartan scheme of rank three. Assume that Rre is a finite irreducible root system of type C. Then all entries of the Cartan matrices of C are greater or equal to −7. Proof. It can be assumed that C is connected. We prove the theorem indirectly. To do so we may assume that a ∈ A such that ca12 ≤ −8. a Then Proposition 2.6 implies that #(R+ ∩ (Zα1 + Zα2 )) ≥ 5. By Lemma 3.12 there exists k0 ∈ {0, 1, 2, 3, 4} such that α := k0 α1 + 2α2 + a α3 ∈ R+ and α − α1 ∈ / Ra . By Lemma 3.2 and the choice of k0 the set {α, α1 } is a base for V a (α, α1 ) at a. Corollary 2.5 implies that there exists a root γ ∈ Ra such that {α, α1 , γ} is a base for ZI at a. Let d ∈ A, w ∈ Hom(a, d), and i1 , i2 , i3 ∈ I such that w(α) = αi1 , w(α1) = αi2 , w(γ) = αi3 . Let b = ρ1 (a). Again by Lemma 3.12 there exists k1 ∈ {0, 1, 2, 3, 4} b such that β := k1 α1 + 2α2 + α3 ∈ R+ . Thus a R+ ∋ σ1b (β) = (−k1 − 2ca12 − ca13 )α1 + 2α2 + α3 .

Further, −k1 − 2ca12 − ca13 − k0 > −ca12 since k0 ≤ 2 if ca13 = 0. Hence αi1 + (1 − ca12 )αi2 ∈ Rd , that is, cdi2 i1 < ca12 ≤ −8. We conclude that there exists no lower bound for the entries of the Cartan matrices of C, which is a contradiction to the finiteness of Rre (C). This proves the theorem.  Remark 3.14. The bound in the theorem is not sharp. After completing the classification one can check that the entries of the Cartan matrices of C are always at least −6. The entry −6 appears for example in the Cartan scheme corresponding to the root system with number 53, see Corollary 2.9. Proposition 3.15. Let C be an irreducible connected simply connected Cartan scheme of rank three. Assume that Rre (C) is a finite root system of type C. Let e be the number of vertices, k the number of edges, and f the number of (2-dimensional) faces of the object change diagram of C. Then e − k + f = 2. Proof. Vertices of the object change diagram correspond to elements of A. Since C is connected and Rre (C) is finite, the set A is finite. Consider the equivalence relation on I × A, where (i, a) is equivalent to (j, b) for some i, j ∈ I and a, b ∈ A, if and only if i = j and b ∈ {a, ρi (a)}. (This is also known as the pushout of I × A along the

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bijections id : I × A → I × A and ρ : I × A → I × A, (i, a) 7→ (i, ρi (a)).) Since C is simply connected, ρi (a) 6= a for all i ∈ I and a ∈ A. Then edges of the object change diagram correspond to equivalence classes in I ×A. Faces of the object change diagram can be defined as equivalence classes of triples (i, j, a) ∈ I × I × A \ {(i, i, a) | i ∈ I, a ∈ A}, where (i, j, a) and (i′ , j ′, b) are equivalent for some i, j, i′ , j ′ ∈ I and a, b ∈ A if and only if {i, j} = {i′ , j ′} and b ∈ {(ρj ρi )m (a), ρi (ρj ρi )m (a) | m ∈ N0 }. Since C is simply connected, (R4) implies that the face corresponding to a triple (i, j, a) is a polygon with 2mai,j vertices. For each face choose a triangulation by non-intersecting diagonals. Let d be the total number of diagonals arising this way. Now consider the following two-dimensional simplicial complex C: The 0-simplices are the objects. The 1-simplices are the edges and the chosen diagonals of the faces of the object change diagram. The 2-simplices are the f +d triangles. Clearly, each edge is contained in precisely two triangles. By [tD91, Ch. III, (3.3), 2,3] the geometric realization X of C is a closed 2-dimensional surface without boundary. The space X is connected and compact. Any two morphisms of W(C) with same source and target are equal because C is simply connected. By [CH09c, Thm. 2.6] this equality follows from the Coxeter relations. A Coxeter relation means for the object change diagram that for the corresponding face and vertex the two paths along the sides of the face towards the opposite vertex yield the same morphism. Hence diagonals can be interpreted as representatives of paths in a face between two vertices, and then all loops in C become homotopic to the trivial loop. Hence X is simply connected and therefore homeomorphic to a two-dimensional sphere by [tD91, Ch. III, Satz 6.9]. Its Euler characteristic is 2 = e − (k + d) + (f + d) = e − k + f .  Remark 3.16. Assume that C is connected and simply connected, and let a ∈ A. Then any pair of opposite 2-dimensional faces of the object change diagram can be interpreed as a plane in ZI containing at least a . Indeed, let b ∈ A and i1 , i2 ∈ I with two positive roots α, β ∈ R+ i1 6= i2 . Since C is connected and simply connected, there exists a unique w ∈ Hom(a, b). Then V a (w −1(αi1 ), w −1(αi2 )) is a plane in ZI containing at least two positive roots. One can easily check that this plane is independent of the choice of the representative of the face determined by (i1 , i2 , b) ∈ I × I × A. Further, let w0 ∈ Hom(b, d), where d ∈ A, be the longest element in Hom(b, W(C)). Let j1 , j2 ∈ I such that w0 (αin ) = −αjn for n = 1, 2. Then (j1 , j2 , d) determines the plane V a ((w0 w)−1 (αj1 ), (w0 w)−1(αj2 )) = V a (w −1(αi1 ), w −1 (αi2 )).

FINITE WEYL GROUPOIDS OF RANK THREE

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Figure 2. The object change diagram of the last root system of rank three

This way we attached to any pair of (2-dimensional) opposite faces of the object change diagram a plane containing at least two positive roots. Let < be a semigroup ordering on ZI such that 0 < γ for all a a γ ∈ R+ . Let α, β ∈ R+ with α 6= β, and assume that α and β are the a smallest elements in R+ ∩ V a (α, β) with respect to 2 for all V ∈ M, then V ∈M #(V ∩ R+ ) ≥ 3#M contradicting Thm. 3.17. Hence for all objects a there exists a plane a V with #(V ∩ R+ ) = 2. Now consider the object change diagram and count the number of faces: Let 2qi be the number of faces with 2i edges. Then Thm. 3.17 translates to X X (3.5) iqi = −3 + 3 qi . i≥2

i≥2

FINITE WEYL GROUPOIDS OF RANK THREE

21

Assume that there exists no object adjacent to a square and a hexagon. Since Rre (C) is irreducible, no two squares have a common edge, see Lemma 3.12(1). Look at the edges ending in vertices of squares, and count each edge once for both polygons adjacent to it. One checks that there are at least twice as many edges adjacent to a polygon with at least 8 vertices as edges of squares. This gives that X 2i · 2qi ≥ 2 · 4 · 2q2 . i≥4

P By Equation (3.5) we then have −3 + 3 i≥2 qi ≥ 4q2 + 2q2 + 3q3 , that P is, q2 < i≥4 qi . But then in average each face has more than 6 edges which contradicts Thm. 3.17. Hence there is an object a such that a there exist α, β, γ ∈ R+ as above satisfying Equation (3.4). We have a α+γ, β+γ, α+β+γ ∈ R+ by Lemma 3.12(1),(2) and Corollary 2.9.  4. The classification In this section we explain the classification of connected simply connected Cartan schemes of rank three such that Rre (C) is a finite irreducible root system. We formulate the main result in Theorem 4.1. The proof of Theorem 4.1 is performed using computer calculations based on results of the previous sections. Our algorithm described below is sufficiently powerful: The implementation in C terminates within a few hours on a usual computer. Removing any of the theorems, the calculations would take at least several weeks. Theorem 4.1. (1) Let C be a connected Cartan scheme of rank three with I = {1, 2, 3}. Assume that Rre (C) is a finite irreducible root system of type C. Then there exists an object a ∈ A and a linear map τ ∈ a ) is one Aut(ZI ) such that τ (αi ) ∈ {α1 , α2 , α3 } for all i ∈ I and τ (R+ a of the sets listed in Appendix A. Moreover, τ (R+ ) with this property is uniquely determined. (2) Let R be one of the 55 subsets of Z3 appearing in Appendix A. There exists up to equivalence a unique connected simply connected Cartan scheme C(I, A, (ρi )i∈I , (C a )a∈A ) such that R ∪ −R is the set of real roots Ra in an object a ∈ A. Moreover Rre (C) is a finite irreducible root system of type C. Let < be the lexicographic ordering on Z3 such that α3 < α2 < α1 . Then α > 0 for any α ∈ N30 \ {0}. Let C be a connected Cartan scheme with I = {1, 2, 3}. Assume that Rre (C) is a finite irreducible root system of type C. Let a ∈ A. a By Theorem 2.10 we may construct R+ inductively by starting with a R+ = {α3 , α2 , α1 }, and appending in each step a sum of a pair of

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a positive roots which is greater than all roots in R+ we already have. During this process, we keep track of all planes containing at least two positive roots, and the positive roots on them. Lemma 3.3 implies that for any known root α and new root β either V a (α, β) contains no other known positive roots, or β is not part of the unique base for V a (α, β) at a consisting of positive roots. In the first case the roots α, β generate a new plane. It can happen that Vol2 (α, β) > 1, and then {α, β} is not a base for V a (α, β) at a, but we don’t care about that. In the second a case the known roots in V a (α, β) ∩ R+ together with β have to form an F -sequence by Proposition 2.6. Sometimes, by some theorem (see the details below) we know that it is not possible to add more positive roots to a plane. Then we can mark it as “finished”. Remark that to obtain a finite number of root systems as output, we have to ensure that we compute only irreducible systems since there are infinitely many inequivalent reducible root systems of rank two. Hence starting with {α3 , α2 , α1 } will not work. However, by Corollary 3.18, starting with {α3 , α2 , α2 + α3 , α1 , α1 + α2 , α1 + α2 + α3 } will still yield at least one root system for each desired Cartan scheme (notice that any roots one would want to add are lexicographically greater). In this section, we will call root system fragment (or rsf) the following set of data associated to a set of positive roots R in construction:

• • • • •

normal vectors for the planes with at least two positive roots labels of positive roots on these planes Cartan entries corresponding to the root systems of the planes an array of flags for finished planes the sum sR of #(V ∩ R) over all planes V with at least two positive roots, see Theorem 3.17 • for each root r ∈ R the list of planes it belongs to.

These data can be obtained directly from R, but the calculation is faster if we continuously update them. We divide the algorithm into three parts. The main part is Algorithm 4.4, see below. The first part updates a root system fragment to a new root and uses Theorems 3.10 and 3.13 to possibly refuse doing so: ˜ α) Algorithm 4.2. AppendRoot(α,B,B,ˆ Append a root to an rsf. ˜ a root α. Input: a root α, an rsf B, an empty rsf B, ˆ

FINITE WEYL GROUPOIDS OF RANK THREE

23

 ˜ 0 : if α may be appended, new rsf is then in B,    1 : if α may not be appended, Output: a a 2 : if α ∈ R+ implies the existence of β ∈ R+    with α ˆ < β < α. 1. Let r be the number of planes containing at least two elements of R. For documentation purposes let V1 , . . . , Vr denote these planes. For any i ∈ {1, . . . , r} let vi be a normal vector for Vi , and let Ri be the F -sequence of Vi ∩R. Set i ← 1, g ← 1, c ← [], p ← [ ], d ← { }. During the algorithm c will be an ordered subset of {1, . . . , r}, p a corresponding list of “positions”, and d a subset of R. 2. If i ≤ r and g 6= 0, then compute the scalar product g := (α, vi). (Then g = det(α, γ1, γ2 ) = ±Vol3 (α, γ1, γ2 ), where {γ1 , γ2} is the basis of Vi consisting of positive roots.) Otherwise go to Step 6. 3. If g = 0 then do the following: If Vi is not finished yet, then check if α extends Ri to a new F -sequence. If yes, add the roots of Ri to d, append i to c, append the position of the insertion of α in Ri to p, let g ← 1, and go to Step 5. 4. If g 2 = 1, then use Corollary 3.11: Let γ1 and γ2 be the beginning and the end of the F -sequence Ri , respectively. (Then {γ1 , γ2 } is a base for Vi at a). Let δ1 ← α − γ1 , δ2 ← α − γ2 . If δ1 , δ2 ∈ / R, then return 1 if δ1 , δ2 ≤ α ˆ and return 2 otherwise. 5. Set i ← i + 1 and go to Step 2. 6. If there is no objection appending α so far, i.e. g 6= 0, then copy ˜ and include α into B: ˜ use c, p to extend existing F B to B sequences, and use (the complement of) d to create new planes. Finally, apply Theorem 3.13: If there is a Cartan entry lesser than −7 then return 1, else return 0. If g = 0 then return 2.

The second part looks for small roots which we must include in any case. The function is based on Proposition 3.6. This is a strong restriction during the process. Algorithm 4.3. RequiredRoot(R,B,ˆ α) Find a smallest required root under the assumption that all roots ≤ α ˆ are known. Input: R a set of roots, B an rsf for R, α ˆ a root.

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  if we cannot determine such a root, 0 Output: 1, ε if we have found a small missing root ε with ε > α, ˆ  2 if the given configuration is impossible.

1. Initialize the return value f ← 0. 2. We use the same notation as in Algo. 4.2, step 1. For all γ1 in R and all (j, k) ∈ {1, . . . , r} × {1, . . . , r} such that j 6= k, γ1 ∈ Rj ∩ Rk , and both Rj , Rk contain two elements, let γ2 , γ3 ∈ R such that Rj = {γ1 , γ2 }, Rk = {γ1 , γ3}. If Vol3 (γ1 , γ2, γ3 ) = 1, then do Steps 3 to 6. 3. ξ2 ← γ1 + γ2 , ξ3 ← γ1 + γ3 . 4. If α ˆ ≥ ξ2 : If α ˆ ≥ ξ3 or plane Vk is already finished, then return 2. If f = 0 or ε > ξ3 , then ε ← ξ3 , f ← 1. Go to Step 2 and continue loop. 5. If α ˆ ≥ ξ3 : If plane Vj is already finished, then return 2. If f = 0 or ε > ξ2 , then ε ← ξ2 , f ← 1. 6. Go to Step 2 and continue loop. 7. Return f, ε.

Finally, we resursively add roots to a set, update the rsf and include required roots: Algorithm 4.4. CompleteRootSystem(R,B,ˆ α,u,β) Collects potential new roots, appends them and calls itself again. Input: R a set of roots, B an rsf for R, α ˆ a lower bound for new roots, u a flag, β a vector which is necessarily a root if u = True. Output: Root systems containing R. 1. Check Theorem 3.17: If sR = 3(r − 1), where r is the number of planes containing at least two positive roots, then output R (and continue). We have found a potential root system. 2. If we have no required root yet, i.e. u = False, then f, ε :=RequiredRoot(R, B, α). ˆ If f = 1, then we have found a required root; we call CompleteRootSystem(R, B, α, ˆ T rue, ε) and terminate. If f = 2, then terminate. ˜ will be the 3. Potential new roots will be collected in Y ← { }; B new rsf. 4. For all planes Vi of B which are not finished, do Steps 5 to 7. 5. ν ← 0. 6. For ζ in the set of roots that may be added to the plane Vi such that ζ > α, ˆ do the following: • set ν ← ν + 1.

FINITE WEYL GROUPOIDS OF RANK THREE

25

• If ζ ∈ / Y , then Y ← Y ∪ {ζ}. If moreover u = False or β > ζ, then ˜ α – y ← AppendRoot(ζ, B, B, ˆ ); ˜ ζ, u, β). – if y = 0 then CompleteRootSystem(R∪{ζ}, B, – if y = 1 then ν ← ν − 1. ˜ 7. If ν = 0, then mark Vi as finished in B. ˜ α) 8. if u = True and AppendRoot(β, B, B, ˆ = 0, then call ˜ CompleteRootSystem(R ∪ {β}, B, β, False, β). Terminate the function call. Note that we only used necessary conditions for root systems, so after the computation we still need to check which of the sets are indeed root systems. A short program in Magma confirms that Algorithm 4.4 yields only root systems, for instance using this algorithm: Algorithm 4.5. RootSystemsForAllObjects(R) a Returns the root systems for all objects if R = R+ determines a Cartan re scheme C such that R (C) is an irreducible root system. Input: R the set of positive roots at one object. Output: the set of root systems at all objects, or {} if R does not yield a Cartan scheme as desired. 1. N ← [R], M ← {}. 2. While |N| > 0, do steps 3 to 5. 3. Let F be the last element of N. Remove F from N and include it to M. 4. Let C be the Cartan matrix of F . Compute the three simple reflections given by C. 5. For each simple reflection s, do: • Compute G := {s(v) | v ∈ F }. If an element of G has positive and negative coefficients, then return {}. Otherwise mutliply the negative roots of G by −1. • If G ∈ / M, then append G to N. 6. Return M. We list all 55 root systems in Appendix A. It is also interesting to summarize some of the invariants, which is done in Table 1. Let O = {Ra | a ∈ A} denote the set of different root systems. By identifying objects with the same root system one obtains a quotient Cartan scheme of the simply connected Cartan scheme of the classification. In the fifth column we give the automorphism group of one (equivalently, any) object of this quotient. The last column gives the multiplicities

26

M. CUNTZ AND I. HECKENBERGER

of planes; for example 37 means that there are 7 different planes containing precisely 3 positive roots. Nr. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

a |R+ | 6 7 8 9 9 10 10 11 12 12 13 13 13 13 14 15 16 16 17 17 17 18 18 19 19 19 19 19 20 20 20 21 21 21 22 25 25 25

|O| 1 4 5 1 1 5 10 9 21 14 4 12 2 2 56 16 36 24 10 10 10 30 90 25 8 50 25 8 27 110 110 15 30 5 44 42 14 28

|A| 24 32 40 48 48 60 60 72 84 84 96 96 96 96 112 128 144 144 160 160 160 180 180 200 192 200 200 192 216 220 220 240 240 240 264 336 336 336

Hom(a) A3 A1 × A1 × A1 B2 B3 B3 A1 × A2 A2 A1 × A1 × A1 A1 × A1 A2 G2 × A1 A1 × A1 × A1 B3 B3 A1 A1 × A1 × A1 A1 × A1 A2 B2 × A1 B2 × A1 B2 × A1 A2 A1 A1 × A1 × A1 G2 × A1 A1 × A1 A1 × A1 × A1 G2 × A1 B2 A1 A1 B2 × A1 A1 × A1 × A1 B3 A2 A1 × A1 × A1 G2 × A1 A1 × A2

planes 23 , 34 , 23 , 36 , 24 , 36 , 41 , 26 , 34 , 43 , 26 , 34 , 43 , 26 , 37 , 43 , 26 , 37 , 43 , 27 , 38 , 44 , 28 , 310 , 43 , 51 , 29 , 37 , 46 , 29 , 312 , 43 , 61 , 210 , 310 , 43 , 52 , 212 , 34 , 49 , 212 , 34 , 49 , 211 , 312 , 44 , 52 , 213 , 312 , 46 , 52 , 214 , 315 , 46 , 51 , 61 , 215 , 313 , 46 , 53 , 216 , 316 , 47 , 62 , 216 , 316 , 47 , 62 , 218 , 312 , 47 , 54 , 218 , 318 , 46 , 53 , 61 , 219 , 316 , 46 , 55 , 220 , 320 , 46 , 54 , 61 , 221 , 318 , 46 , 64 , 220 , 320 , 46 , 54 , 61 , 220 , 320 , 46 , 54 , 61 , 224 , 312 , 46 , 56 , 61 , 220 , 326 , 44 , 54 , 81 , 221 , 324 , 46 , 54 , 71 , 223 , 320 , 47 , 55 , 61 , 222 , 328 , 46 , 54 , 81 , 226 , 320 , 49 , 54 , 62 , 224 , 324 , 49 , 64 , 227 , 325 , 49 , 53 , 63 , 233 , 334 , 412 , 52 , 63 , 81 , 236 , 330 , 49 , 56 , 64 , 236 , 330 , 49 , 56 , 64 ,

FINITE WEYL GROUPOIDS OF RANK THREE

Nr. 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

a |R+ | 25 26 26 27 27 27 28 28 28 29 29 29 30 31 31 34 37

|O| 7 182 182 49 98 98 420 210 70 56 112 112 238 21 21 102 15

|A| 336 364 364 392 392 392 420 420 420 448 448 448 476 504 504 612 720

Hom(a) B3 A1 A1 A1 × A1 × A1 A1 × A1 A1 × A1 1 A1 A2 A1 × A1 × A1 A1 × A1 A1 × A1 A1 G2 × A1 G2 × A1 A2 B3

27

planes 236 , 328 , 415 , 66 , 235 , 339 , 410 , 54 , 63 , 81 , 237 , 336 , 49 , 56 , 63 , 71 , 238 , 342 , 49 , 56 , 63 , 81 , 239 , 340 , 410 , 56 , 62 , 72 , 239 , 340 , 410 , 56 , 62 , 72 , 241 , 344 , 411 , 56 , 62 , 71 , 81 , 242 , 342 , 412 , 56 , 61 , 73 , 242 , 342 , 412 , 56 , 61 , 73 , 244 , 346 , 413 , 56 , 62 , 82 , 245 , 344 , 414 , 56 , 61 , 72 , 81 , 245 , 344 , 414 , 56 , 61 , 72 , 81 , 249 , 344 , 417 , 56 , 61 , 71 , 82 , 254 , 342 , 421 , 56 , 61 , 83 , 254 , 342 , 421 , 56 , 61 , 83 , 260 , 363 , 418 , 56 , 64 , 83 , 272 , 372 , 424 , 610 , 83 ,

Table 1: Invariants of irreducible root systems of rank three

At first sight, one is tempted to look for a formula for the number of objects in the universal covering depending on the number of roots. There is an obvious one: consider the coefficients of 4/((1−x)2 (1−x4 )). However, there are exceptions, for example nr. 29 with 20 positive roots and 216 objects (instead of 220). Rank 3 Nichols algebras of diagonal type with finite irreducible arithmetic root system are classified in [Hec05, Table 2]. In Table 2 we identify the Weyl groupoids of these Nichols algebras. row in [Hec05, Table 2] 1 Weyl groupoid 1 row in [Hec05, Table 2] 10 Weyl groupoid 2

2 5 11 2

3 4 12 5

4 1 13 13

5 5 14 5

6 3 15 6

7 11 16 7

8 1 17 8

9 2 18 14

Table 2: Weyl groupoids of rank three Nichols algebras of diagonal type

Appendix A. Irreducible root systems of rank three We give the roots in a multiplicative notation to save space: The word 1x 2y 3z corresponds to xα3 + yα2 + zα1 . Notice that we have chosen a “canonical” object for each groupoid. a a Write π(R+ ) for the set R+ where the coordinates are permuted via a ) | a ∈ π ∈ S3 . Then the set listed below is the minimum of {π(R+

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A, π ∈ S3 } with respect to the lexicographical ordering on the sorted sequences of roots. Nr. 1 with 6 positive roots: 1, 2, 3, 12, 13, 123 Nr. 2 with 7 positive roots: 1, 2, 3, 12, 13, 23, 123 Nr. 3 with 8 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 12 23 Nr. 4 with 9 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 12 23, 12 232 Nr. 5 with 9 positive roots: 1, 2, 3, 12, 23, 12 2, 123, 12 23, 12 22 3 Nr. 6 with 10 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 12 23, 13 23 Nr. 7 with 10 positive roots: 1, 2, 3, 12, 13, 23, 12 2, 123, 12 23, 12 22 3 Nr. 8 with 11 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 12 23, 13 23, 13 22 3 Nr. 9 with 12 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 13 2, 12 23, 13 23, 13 22 3, 14 22 3 Nr. 10 with 12 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 12 23, 13 23, 12 22 3, 13 22 3 Nr. 11 with 13 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 13 2, 12 23, 13 22 , 13 23, 13 22 3, 14 22 3 Nr. 12 with 13 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 23, 14 23, 14 22 3 Nr. 13 with 13 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 12 23, 13 23, 12 22 3, 13 22 3, 14 22 3 Nr. 14 with 13 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 132 , 12 23, 1232 , 12 232 , 13 232 , 13 22 32 Nr. 15 with 14 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 23, 14 23, 13 22 3, 14 22 3 Nr. 16 with 15 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 23, 14 23, 13 22 3, 14 22 3, 15 22 3 Nr. 17 with 16 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 22 , 13 23, 14 23, 13 22 3, 14 22 3, 15 22 3 Nr. 18 with 16 positive roots: 1, 2, 3, 12, 23, 12 2, 123, 13 2, 12 23, 122 3, 13 23, 12 22 3, 13 22 3, 14 22 3, 14 23 3, 14 23 32 Nr. 19 with 17 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 23, 14 23, 15 23, 14 22 3, 15 22 3, 16 22 3 Nr. 20 with 17 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 22 , 13 23, 14 23, 13 22 3, 14 22 3, 15 22 3, 15 22 32 Nr. 21 with 17 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 13 2, 12 23, 13 23, 12 22 3, 13 22 3, 14 22 3, 15 22 3, 15 23 3, 15 23 32 , 16 23 32 Nr. 22 with 18 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 22 , 13 23, 14 23, 13 22 3, 14 22 3, 15 22 3, 15 23 3, 16 23 3 Nr. 23 with 18 positive roots: 1, 2, 3, 12, 13, 23, 12 2, 123, 13 2, 12 23, 122 3, 13 23, 12 22 3, 13 22 3, 14 22 3, 13 23 3, 14 23 3, 14 23 32 Nr. 24 with 19 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 13 2, 12 23, 14 2, 13 23, 14 23, 13 22 3, 14 22 3, 15 22 3, 16 22 3, 16 23 3, 17 23 3, 17 23 32 Nr. 25 with 19 positive roots: 1, 2, 3, 12, 23, 12 2, 123, 13 2, 12 23, 14 2, 13 23, 12 22 3, 14 23, 13 22 3, 14 22 3, 15 22 3, 16 22 3, 16 23 3, 16 23 32 Nr. 26 with 19 positive roots: 1, 2, 3, 12, 13, 23, 12 2, 122 , 123, 13 2, 12 23, 122 3, 13 23, 12 22 3, 13 22 3, 14 22 3, 13 23 3, 14 23 3, 14 23 32

FINITE WEYL GROUPOIDS OF RANK THREE

29

Nr. 27 with 19 positive roots: 1, 2, 3, 12, 13, 23, 12 2, 123, 13 2, 12 23, 122 3, 13 22 , 13 23, 12 22 3, 13 22 3, 14 22 3, 13 23 3, 14 23 3, 14 23 32 Nr. 28 with 19 positive roots: 1, 2, 3, 12, 23, 12 2, 123, 13 2, 12 23, 13 22 , 13 23, 12 22 3, 13 22 3, 14 22 3, 13 23 3, 14 23 3, 15 23 3, 16 23 3, 16 24 3 Nr. 29 with 20 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 14 22 3, 15 22 3, 16 22 3, 16 23 3, 17 23 3 Nr. 30 with 20 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 14 22 3, 15 22 3, 16 22 3, 16 23 3, 17 23 3, 17 23 32 Nr. 31 with 20 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 22 , 13 23, 12 22 3, 14 23, 13 22 3, 14 22 3, 15 22 3, 14 23 3, 15 23 3, 16 23 32 Nr. 32 with 21 positive roots: 1, 2, 3, 12, 13, 12 2, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 14 22 3, 15 22 3, 16 22 3, 16 23 3, 17 23 3, 17 23 32 Nr. 33 with 21 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 22 , 13 23, 12 22 3, 14 23, 13 22 3, 14 22 3, 15 22 3, 14 23 3, 15 23 3, 16 23 3, 16 23 32 Nr. 34 with 21 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 22 , 13 23, 14 23, 13 22 3, 14 22 3, 15 22 3, 15 23 3, 15 22 32 , 16 23 3, 16 23 32 , 17 23 32 Nr. 35 with 22 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 22 , 13 23, 12 22 3, 14 23, 13 22 3, 14 22 3, 15 22 3, 14 23 3, 15 23 3, 15 22 32 , 15 23 32 , 16 23 32 Nr. 36 with 25 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 17 22 3, 16 23 3, 17 23 3, 18 23 3, 18 23 32 Nr. 37 with 25 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 23, 14 23, 13 22 3, 15 23, 14 22 3, 15 22 3, 16 22 3, 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 Nr. 38 with 25 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 122 , 123, 13 2, 12 23, 122 3, 13 23, 12 22 3, 14 23, 13 22 3, 14 22 3, 13 23 3, 13 22 32 , 14 23 3, 15 23 3, 14 23 32 , 15 23 32 , 16 23 32 , 17 24 32 Nr. 39 with 25 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 13 22 , 13 23, 12 22 3, 14 23, 13 22 3, 14 22 3, 15 22 3, 14 23 3, 15 23 3, 15 22 32 , 16 23 3, 15 23 32 , 16 23 32 , 17 23 32 , 17 24 32 Nr. 40 with 26 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 Nr. 41 with 26 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 23, 14 22 3, 15 22 3, 16 22 3, 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 Nr. 42 with 27 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 Nr. 43 with 27 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 23, 14 22 3, 15 22 3, 16 22 3, 15 22 32 , 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 Nr. 44 with 27 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 23, 14 22 3, 15 22 3, 16 22 3, 17 22 3, 16 23 3, 17 23 3, 17 22 32 , 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 Nr. 45 with 28 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 15 22 32 , 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 Nr. 46 with 28 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 23, 14 22 3, 15 22 3, 16 22 3,

30

M. CUNTZ AND I. HECKENBERGER

15 22 32 , 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 , 19 24 32 Nr. 47 with 28 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 23, 14 22 3, 15 22 3, 16 22 3, 15 22 32 , 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 , 111 24 32 Nr. 48 with 29 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 15 22 32 , 17 22 3, 16 23 3, 17 23 3, 17 22 32 , 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 Nr. 49 with 29 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 15 22 32 , 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 , 19 24 32 Nr. 50 with 29 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 15 22 32 , 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 , 111 24 32 Nr. 51 with 30 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 15 22 32 , 17 22 3, 16 23 3, 17 23 3, 17 22 32 , 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 , 19 24 32 Nr. 52 with 31 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 23, 15 2, 14 23, 16 2, 15 23, 14 22 3, 16 23, 15 22 3, 17 23, 16 22 3, 17 22 3, 18 22 3, 19 22 3, 110 22 3, 19 23 3, 110 23 3, 111 23 3, 110 23 32 , 111 23 32 , 112 23 32 Nr. 53 with 31 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 15 22 32 , 17 22 3, 16 23 3, 17 23 3, 17 22 32 , 18 23 3, 17 23 32 , 18 23 32 , 19 23 32 , 19 24 32 , 111 24 32 Nr. 54 with 34 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 15 23 3, 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 18 24 3, 18 23 32 , 19 24 3, 19 23 32 , 19 24 32 , 111 24 32 , 111 25 32 , 112 25 32 Nr. 55 with 37 positive roots: 1, 2, 3, 12, 13, 12 2, 12 3, 123, 13 2, 12 23, 14 2, 13 22 , 13 23, 14 23, 13 22 3, 15 22 , 15 23, 14 22 3, 15 22 3, 16 22 3, 15 23 3, 17 22 3, 16 23 3, 17 23 3, 18 23 3, 17 23 32 , 19 23 3, 18 24 3, 18 23 32 , 19 24 3, 19 23 32 , 110 24 3, 19 24 32 , 111 24 32 , 111 25 32 , 112 25 32 , 113 25 32

References ´ ements de [Bou68] N. Bourbaki, Groupes et alg`ebres de Lie, ch. 4, 5 et 6, El´ math´ematique, Hermann, Paris, 1968. [CH09a] M. Cuntz and I. Heckenberger, Reflection groupoids of rank two and cluster algebras of type A, Preprint arXiv:0911.3051v1 (2009), 18 pp. [CH09b] , Weyl groupoids of rank two and continued fractions, Algebra and Number Theory (2009), 317–340. [CH09c] , Weyl groupoids with at most three objects, J. Pure Appl. Algebra 213 (2009), 1112–1128. [Gr¨ u09] B. Gr¨ unbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Math. Contemp. 2 (2009), 1–25. [Hec05] I. Heckenberger, Classification of arithmetic root systems of rank 3, Ac´ tas del “XVI Coloquio Latinoamericano de Algebra” (Colonia, Uruguay, 2005), 2005, pp. 227–252. , The Weyl groupoid of a Nichols algebra of diagonal type, Invent. [Hec06] Math. 164 (2006), 175–188. , Rank 2 Nichols algebras with finite arithmetic root system, Algebr. [Hec08] Represent. Theory 11 (2008), 115–132. [HY08] I. Heckenberger and H. Yamane, A generalization of Coxeter groups, root systems, and Matsumoto’s theorem, Math. Z. 259 (2008), 255–276. [Kac77] V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96. , Infinite dimensional Lie algebras, Cambridge Univ. Press, 1990. [Kac90]

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¨ [Mel41] E. Melchior, Uber Vielseite der projektiven Ebene, Deutsche Mathematik 5 (1941), 461–475. [tD91] T. tom Dieck, Topologie, de Gruyter Lehrbuch, Walter de Gruyter & Co., Berlin, 1991. ¨t KaiserslauMichael Cuntz, Fachbereich Mathematik, Universita tern, Postfach 3049, D-67653 Kaiserslautern, Germany E-mail address: [email protected] ´n Heckenberger, Philipps-Universita ¨t Marburg, Fachbereich Istva Mathematik und Informatik, Hans-Meerwein-Straße, D-35032 Marburg, Germany E-mail address: [email protected]